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Let $G$ be a one-relator group $\langle A \mid R = 1 \rangle$. Then clearly $G$ is finite if and only if it is cyclic of finite order, i.e. can be given by a presentation $\langle a \mid a^n = 1 \rang …

Let $d$ be a square-free positive integer, and let $\mathcal{O}_d$ be the ring of integers of the quadratic imaginary number field $\mathbb{Q}(\sqrt{-d})$. Consider the Bianchi group $\Gamma_d = \oper …

As indicated in the comments, it's undecidable in general to take as input a finite presentation of a group and try to output whether or not this group is free or not. This is a direct consequence of …

Here is a very classical example. As stated in the comments, Gromov was an early proponent of importing ideas from geometry to group theory, but already thirty years earlier there was work in this dir …

Let $G$ be a finitely presented simple group. By Kuznetsov (1958), $G$ has decidable word problem. However, by Scott [1], $G$ may have undecidable conjugacy problem. Is anything known about other deci …

After some digging, I was able to find that the answer to my question exists: the problem can be undecidable. Rattaggi, in an unpublished manuscript (available here), proved that there exists a finite …

Polycyclic groups are LERF, by Mal'cev 1948. In particular, all nilpotent and all abelian groups are LERF. As mentioned in the comments, as not all one-relator groups are residually finite, not all on …

Here is a connection that I recently noticed, but I haven't quite been able to make sense of. It might follow from well-known facts; apologies, if so. This is quite far from my area. First, in "Yang-M …

I'll write out the answer to your first question (solving the word problem in one-relator groups) for how one might do this in practice. I'll focus on the case $\ell = 2, m=3, p=2$, but you'll hopeful …

Here's a few examples in line with classical combinatorial group theory. Though small cancellation groups as a whole have already been mentioned, one important subclass of these are the one-relator g …

Given two metric spaces $(M_1, d_1)$ and $(M_2, d_2)$, a map $\phi \colon (M_1, d_1) \to (M_2, d_2)$ is a large-scale Lipschitz essentially surjective map if there exist constants $A \geq 1, B \geq 0$ …

They are all residually finite. They are not all subgroup separable/LERF. They do not all have decidable submonoid membership problem. Residual finiteness is a result which can be found in the (ap …

A preprint by E. Rauzy appeared today on the arXiv, and gives a negative answer to this question. In other words (if the proof is correct), there exists a f.g. residually finite group with decidable w …

Burger and Mozes constructed (Burger and Mozes - Lattices in products of trees) infinite, finitely presented, torsion-free simple groups which split as amalgams of two finitely generated free groups o …

While most other answers have mentioned computational problems related to finitely presented (but generally infinite) groups, there are many problems in finite group theory which are either conjecture …